src/HOL/Transitive_Closure.thy
author paulson
Tue Jun 28 15:27:45 2005 +0200 (2005-06-28)
changeset 16587 b34c8aa657a5
parent 16514 090c6a98c704
child 17589 58eeffd73be1
permissions -rw-r--r--
Constant "If" is now local
     1 (*  Title:      HOL/Transitive_Closure.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1992  University of Cambridge
     5 *)
     6 
     7 header {* Reflexive and Transitive closure of a relation *}
     8 
     9 theory Transitive_Closure
    10 imports Inductive
    11 uses ("../Provers/trancl.ML")
    12 begin
    13 
    14 text {*
    15   @{text rtrancl} is reflexive/transitive closure,
    16   @{text trancl} is transitive closure,
    17   @{text reflcl} is reflexive closure.
    18 
    19   These postfix operators have \emph{maximum priority}, forcing their
    20   operands to be atomic.
    21 *}
    22 
    23 consts
    24   rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^*)" [1000] 999)
    25 
    26 inductive "r^*"
    27   intros
    28     rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*"
    29     rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"
    30 
    31 consts
    32   trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^+)" [1000] 999)
    33 
    34 inductive "r^+"
    35   intros
    36     r_into_trancl [intro, Pure.intro]: "(a, b) : r ==> (a, b) : r^+"
    37     trancl_into_trancl [Pure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a,c) : r^+"
    38 
    39 syntax
    40   "_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^=)" [1000] 999)
    41 translations
    42   "r^=" == "r \<union> Id"
    43 
    44 syntax (xsymbols)
    45   rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>*)" [1000] 999)
    46   trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>+)" [1000] 999)
    47   "_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>=)" [1000] 999)
    48 
    49 syntax (HTML output)
    50   rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>*)" [1000] 999)
    51   trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>+)" [1000] 999)
    52   "_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>=)" [1000] 999)
    53 
    54 
    55 subsection {* Reflexive-transitive closure *}
    56 
    57 lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*"
    58   -- {* @{text rtrancl} of @{text r} contains @{text r} *}
    59   apply (simp only: split_tupled_all)
    60   apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])
    61   done
    62 
    63 lemma rtrancl_mono: "r \<subseteq> s ==> r^* \<subseteq> s^*"
    64   -- {* monotonicity of @{text rtrancl} *}
    65   apply (rule subsetI)
    66   apply (simp only: split_tupled_all)
    67   apply (erule rtrancl.induct)
    68    apply (rule_tac [2] rtrancl_into_rtrancl, blast+)
    69   done
    70 
    71 theorem rtrancl_induct [consumes 1, induct set: rtrancl]:
    72   assumes a: "(a, b) : r^*"
    73     and cases: "P a" "!!y z. [| (a, y) : r^*; (y, z) : r; P y |] ==> P z"
    74   shows "P b"
    75 proof -
    76   from a have "a = a --> P b"
    77     by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+
    78   thus ?thesis by rules
    79 qed
    80 
    81 lemmas rtrancl_induct2 =
    82   rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
    83                  consumes 1, case_names refl step]
    84  
    85 lemma trans_rtrancl: "trans(r^*)"
    86   -- {* transitivity of transitive closure!! -- by induction *}
    87 proof (rule transI)
    88   fix x y z
    89   assume "(x, y) \<in> r\<^sup>*"
    90   assume "(y, z) \<in> r\<^sup>*"
    91   thus "(x, z) \<in> r\<^sup>*" by induct (rules!)+
    92 qed
    93 
    94 lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard]
    95 
    96 lemma rtranclE:
    97   "[| (a::'a,b) : r^*;  (a = b) ==> P;
    98       !!y.[| (a,y) : r^*; (y,b) : r |] ==> P
    99    |] ==> P"
   100   -- {* elimination of @{text rtrancl} -- by induction on a special formula *}
   101 proof -
   102   assume major: "(a::'a,b) : r^*"
   103   case rule_context
   104   show ?thesis
   105     apply (subgoal_tac "(a::'a) = b | (EX y. (a,y) : r^* & (y,b) : r)")
   106      apply (rule_tac [2] major [THEN rtrancl_induct])
   107       prefer 2 apply (blast!)
   108       prefer 2 apply (blast!)
   109     apply (erule asm_rl exE disjE conjE prems)+
   110     done
   111 qed
   112 
   113 lemma converse_rtrancl_into_rtrancl:
   114   "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> r\<^sup>* \<Longrightarrow> (a, c) \<in> r\<^sup>*"
   115   by (rule rtrancl_trans) rules+
   116 
   117 text {*
   118   \medskip More @{term "r^*"} equations and inclusions.
   119 *}
   120 
   121 lemma rtrancl_idemp [simp]: "(r^*)^* = r^*"
   122   apply auto
   123   apply (erule rtrancl_induct)
   124    apply (rule rtrancl_refl)
   125   apply (blast intro: rtrancl_trans)
   126   done
   127 
   128 lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"
   129   apply (rule set_ext)
   130   apply (simp only: split_tupled_all)
   131   apply (blast intro: rtrancl_trans)
   132   done
   133 
   134 lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"
   135 by (drule rtrancl_mono, simp)
   136 
   137 lemma rtrancl_subset: "R \<subseteq> S ==> S \<subseteq> R^* ==> S^* = R^*"
   138   apply (drule rtrancl_mono)
   139   apply (drule rtrancl_mono, simp)
   140   done
   141 
   142 lemma rtrancl_Un_rtrancl: "(R^* \<union> S^*)^* = (R \<union> S)^*"
   143   by (blast intro!: rtrancl_subset intro: r_into_rtrancl rtrancl_mono [THEN subsetD])
   144 
   145 lemma rtrancl_reflcl [simp]: "(R^=)^* = R^*"
   146   by (blast intro!: rtrancl_subset intro: r_into_rtrancl)
   147 
   148 lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*"
   149   apply (rule sym)
   150   apply (rule rtrancl_subset, blast, clarify)
   151   apply (rename_tac a b)
   152   apply (case_tac "a = b", blast)
   153   apply (blast intro!: r_into_rtrancl)
   154   done
   155 
   156 theorem rtrancl_converseD:
   157   assumes r: "(x, y) \<in> (r^-1)^*"
   158   shows "(y, x) \<in> r^*"
   159 proof -
   160   from r show ?thesis
   161     by induct (rules intro: rtrancl_trans dest!: converseD)+
   162 qed
   163 
   164 theorem rtrancl_converseI:
   165   assumes r: "(y, x) \<in> r^*"
   166   shows "(x, y) \<in> (r^-1)^*"
   167 proof -
   168   from r show ?thesis
   169     by induct (rules intro: rtrancl_trans converseI)+
   170 qed
   171 
   172 lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
   173   by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
   174 
   175 theorem converse_rtrancl_induct[consumes 1]:
   176   assumes major: "(a, b) : r^*"
   177     and cases: "P b" "!!y z. [| (y, z) : r; (z, b) : r^*; P z |] ==> P y"
   178   shows "P a"
   179 proof -
   180   from rtrancl_converseI [OF major]
   181   show ?thesis
   182     by induct (rules intro: cases dest!: converseD rtrancl_converseD)+
   183 qed
   184 
   185 lemmas converse_rtrancl_induct2 =
   186   converse_rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
   187                  consumes 1, case_names refl step]
   188 
   189 lemma converse_rtranclE:
   190   "[| (x,z):r^*;
   191       x=z ==> P;
   192       !!y. [| (x,y):r; (y,z):r^* |] ==> P
   193    |] ==> P"
   194 proof -
   195   assume major: "(x,z):r^*"
   196   case rule_context
   197   show ?thesis
   198     apply (subgoal_tac "x = z | (EX y. (x,y) : r & (y,z) : r^*)")
   199      apply (rule_tac [2] major [THEN converse_rtrancl_induct])
   200       prefer 2 apply rules
   201      prefer 2 apply rules
   202     apply (erule asm_rl exE disjE conjE prems)+
   203     done
   204 qed
   205 
   206 ML_setup {*
   207   bind_thm ("converse_rtranclE2", split_rule
   208     (read_instantiate [("x","(xa,xb)"), ("z","(za,zb)")] (thm "converse_rtranclE")));
   209 *}
   210 
   211 lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"
   212   by (blast elim: rtranclE converse_rtranclE
   213     intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
   214 
   215 lemma rtrancl_unfold: "r^* = Id Un (r O r^*)"
   216   by (auto intro: rtrancl_into_rtrancl elim: rtranclE)
   217 
   218 
   219 subsection {* Transitive closure *}
   220 
   221 lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
   222   apply (simp only: split_tupled_all)
   223   apply (erule trancl.induct)
   224   apply (rules dest: subsetD)+
   225   done
   226 
   227 lemma r_into_trancl': "!!p. p : r ==> p : r^+"
   228   by (simp only: split_tupled_all) (erule r_into_trancl)
   229 
   230 text {*
   231   \medskip Conversions between @{text trancl} and @{text rtrancl}.
   232 *}
   233 
   234 lemma trancl_into_rtrancl: "(a, b) \<in> r^+ ==> (a, b) \<in> r^*"
   235   by (erule trancl.induct) rules+
   236 
   237 lemma rtrancl_into_trancl1: assumes r: "(a, b) \<in> r^*"
   238   shows "!!c. (b, c) \<in> r ==> (a, c) \<in> r^+" using r
   239   by induct rules+
   240 
   241 lemma rtrancl_into_trancl2: "[| (a,b) : r;  (b,c) : r^* |]   ==>  (a,c) : r^+"
   242   -- {* intro rule from @{text r} and @{text rtrancl} *}
   243   apply (erule rtranclE, rules)
   244   apply (rule rtrancl_trans [THEN rtrancl_into_trancl1])
   245    apply (assumption | rule r_into_rtrancl)+
   246   done
   247 
   248 lemma trancl_induct [consumes 1, induct set: trancl]:
   249   assumes a: "(a,b) : r^+"
   250   and cases: "!!y. (a, y) : r ==> P y"
   251     "!!y z. (a,y) : r^+ ==> (y, z) : r ==> P y ==> P z"
   252   shows "P b"
   253   -- {* Nice induction rule for @{text trancl} *}
   254 proof -
   255   from a have "a = a --> P b"
   256     by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+
   257   thus ?thesis by rules
   258 qed
   259 
   260 lemma trancl_trans_induct:
   261   "[| (x,y) : r^+;
   262       !!x y. (x,y) : r ==> P x y;
   263       !!x y z. [| (x,y) : r^+; P x y; (y,z) : r^+; P y z |] ==> P x z
   264    |] ==> P x y"
   265   -- {* Another induction rule for trancl, incorporating transitivity *}
   266 proof -
   267   assume major: "(x,y) : r^+"
   268   case rule_context
   269   show ?thesis
   270     by (rules intro: r_into_trancl major [THEN trancl_induct] prems)
   271 qed
   272 
   273 inductive_cases tranclE: "(a, b) : r^+"
   274 
   275 lemma trancl_unfold: "r^+ = r Un (r O r^+)"
   276   by (auto intro: trancl_into_trancl elim: tranclE)
   277 
   278 lemma trans_trancl: "trans(r^+)"
   279   -- {* Transitivity of @{term "r^+"} *}
   280 proof (rule transI)
   281   fix x y z
   282   assume "(x, y) \<in> r^+"
   283   assume "(y, z) \<in> r^+"
   284   thus "(x, z) \<in> r^+" by induct (rules!)+
   285 qed
   286 
   287 lemmas trancl_trans = trans_trancl [THEN transD, standard]
   288 
   289 lemma rtrancl_trancl_trancl: assumes r: "(x, y) \<in> r^*"
   290   shows "!!z. (y, z) \<in> r^+ ==> (x, z) \<in> r^+" using r
   291   by induct (rules intro: trancl_trans)+
   292 
   293 lemma trancl_into_trancl2: "(a, b) \<in> r ==> (b, c) \<in> r^+ ==> (a, c) \<in> r^+"
   294   by (erule transD [OF trans_trancl r_into_trancl])
   295 
   296 lemma trancl_insert:
   297   "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
   298   -- {* primitive recursion for @{text trancl} over finite relations *}
   299   apply (rule equalityI)
   300    apply (rule subsetI)
   301    apply (simp only: split_tupled_all)
   302    apply (erule trancl_induct, blast)
   303    apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl r_into_trancl trancl_trans)
   304   apply (rule subsetI)
   305   apply (blast intro: trancl_mono rtrancl_mono
   306     [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
   307   done
   308 
   309 lemma trancl_converseI: "(x, y) \<in> (r^+)^-1 ==> (x, y) \<in> (r^-1)^+"
   310   apply (drule converseD)
   311   apply (erule trancl.induct)
   312   apply (rules intro: converseI trancl_trans)+
   313   done
   314 
   315 lemma trancl_converseD: "(x, y) \<in> (r^-1)^+ ==> (x, y) \<in> (r^+)^-1"
   316   apply (rule converseI)
   317   apply (erule trancl.induct)
   318   apply (rules dest: converseD intro: trancl_trans)+
   319   done
   320 
   321 lemma trancl_converse: "(r^-1)^+ = (r^+)^-1"
   322   by (fastsimp simp add: split_tupled_all
   323     intro!: trancl_converseI trancl_converseD)
   324 
   325 lemma converse_trancl_induct:
   326   "[| (a,b) : r^+; !!y. (y,b) : r ==> P(y);
   327       !!y z.[| (y,z) : r;  (z,b) : r^+;  P(z) |] ==> P(y) |]
   328     ==> P(a)"
   329 proof -
   330   assume major: "(a,b) : r^+"
   331   case rule_context
   332   show ?thesis
   333     apply (rule major [THEN converseI, THEN trancl_converseI [THEN trancl_induct]])
   334      apply (rule prems)
   335      apply (erule converseD)
   336     apply (blast intro: prems dest!: trancl_converseD)
   337     done
   338 qed
   339 
   340 lemma tranclD: "(x, y) \<in> R^+ ==> EX z. (x, z) \<in> R \<and> (z, y) \<in> R^*"
   341   apply (erule converse_trancl_induct, auto)
   342   apply (blast intro: rtrancl_trans)
   343   done
   344 
   345 lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
   346 by(blast elim: tranclE dest: trancl_into_rtrancl)
   347 
   348 lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"
   349   by (blast dest: r_into_trancl)
   350 
   351 lemma trancl_subset_Sigma_aux:
   352     "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
   353   apply (erule rtrancl_induct, auto)
   354   done
   355 
   356 lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
   357   apply (rule subsetI)
   358   apply (simp only: split_tupled_all)
   359   apply (erule tranclE)
   360   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
   361   done
   362 
   363 lemma reflcl_trancl [simp]: "(r^+)^= = r^*"
   364   apply safe
   365    apply (erule trancl_into_rtrancl)
   366   apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
   367   done
   368 
   369 lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
   370   apply safe
   371    apply (drule trancl_into_rtrancl, simp)
   372   apply (erule rtranclE, safe)
   373    apply (rule r_into_trancl, simp)
   374   apply (rule rtrancl_into_trancl1)
   375    apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
   376   done
   377 
   378 lemma trancl_empty [simp]: "{}^+ = {}"
   379   by (auto elim: trancl_induct)
   380 
   381 lemma rtrancl_empty [simp]: "{}^* = Id"
   382   by (rule subst [OF reflcl_trancl]) simp
   383 
   384 lemma rtranclD: "(a, b) \<in> R^* ==> a = b \<or> a \<noteq> b \<and> (a, b) \<in> R^+"
   385   by (force simp add: reflcl_trancl [symmetric] simp del: reflcl_trancl)
   386 
   387 lemma rtrancl_eq_or_trancl:
   388   "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"
   389   by (fast elim: trancl_into_rtrancl dest: rtranclD)
   390 
   391 text {* @{text Domain} and @{text Range} *}
   392 
   393 lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
   394   by blast
   395 
   396 lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
   397   by blast
   398 
   399 lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
   400   by (rule rtrancl_Un_rtrancl [THEN subst]) fast
   401 
   402 lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
   403   by (blast intro: subsetD [OF rtrancl_Un_subset])
   404 
   405 lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
   406   by (unfold Domain_def) (blast dest: tranclD)
   407 
   408 lemma trancl_range [simp]: "Range (r^+) = Range r"
   409   by (simp add: Range_def trancl_converse [symmetric])
   410 
   411 lemma Not_Domain_rtrancl:
   412     "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
   413   apply auto
   414   by (erule rev_mp, erule rtrancl_induct, auto)
   415 
   416 
   417 text {* More about converse @{text rtrancl} and @{text trancl}, should
   418   be merged with main body. *}
   419 
   420 lemma single_valued_confluent:
   421   "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
   422   \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
   423 apply(erule rtrancl_induct)
   424  apply simp
   425 apply(erule disjE)
   426  apply(blast elim:converse_rtranclE dest:single_valuedD)
   427 apply(blast intro:rtrancl_trans)
   428 done
   429 
   430 lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
   431   by (fast intro: trancl_trans)
   432 
   433 lemma trancl_into_trancl [rule_format]:
   434     "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
   435   apply (erule trancl_induct)
   436    apply (fast intro: r_r_into_trancl)
   437   apply (fast intro: r_r_into_trancl trancl_trans)
   438   done
   439 
   440 lemma trancl_rtrancl_trancl:
   441     "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r\<^sup>* ==> (a, c) \<in> r\<^sup>+"
   442   apply (drule tranclD)
   443   apply (erule exE, erule conjE)
   444   apply (drule rtrancl_trans, assumption)
   445   apply (drule rtrancl_into_trancl2, assumption, assumption)
   446   done
   447 
   448 lemmas transitive_closure_trans [trans] =
   449   r_r_into_trancl trancl_trans rtrancl_trans
   450   trancl_into_trancl trancl_into_trancl2
   451   rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
   452   rtrancl_trancl_trancl trancl_rtrancl_trancl
   453 
   454 declare trancl_into_rtrancl [elim]
   455 
   456 declare rtranclE [cases set: rtrancl]
   457 declare tranclE [cases set: trancl]
   458 
   459 
   460 
   461 
   462 
   463 subsection {* Setup of transitivity reasoner *}
   464 
   465 use "../Provers/trancl.ML";
   466 
   467 ML_setup {*
   468 
   469 structure Trancl_Tac = Trancl_Tac_Fun (
   470   struct
   471     val r_into_trancl = thm "r_into_trancl";
   472     val trancl_trans  = thm "trancl_trans";
   473     val rtrancl_refl = thm "rtrancl_refl";
   474     val r_into_rtrancl = thm "r_into_rtrancl";
   475     val trancl_into_rtrancl = thm "trancl_into_rtrancl";
   476     val rtrancl_trancl_trancl = thm "rtrancl_trancl_trancl";
   477     val trancl_rtrancl_trancl = thm "trancl_rtrancl_trancl";
   478     val rtrancl_trans = thm "rtrancl_trans";
   479 
   480   fun decomp (Trueprop $ t) = 
   481     let fun dec (Const ("op :", _) $ (Const ("Pair", _) $ a $ b) $ rel ) = 
   482 	let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*")
   483 	      | decr (Const ("Transitive_Closure.trancl", _ ) $ r)  = (r,"r+")
   484 	      | decr r = (r,"r");
   485 	    val (rel,r) = decr rel;
   486 	in SOME (a,b,rel,r) end
   487       | dec _ =  NONE 
   488     in dec t end;
   489   
   490   end); (* struct *)
   491 
   492 simpset_ref() := simpset ()
   493     addSolver (mk_solver "Trancl" (fn _ => Trancl_Tac.trancl_tac))
   494     addSolver (mk_solver "Rtrancl" (fn _ => Trancl_Tac.rtrancl_tac));
   495 
   496 *}
   497 
   498 (* Optional methods
   499 
   500 method_setup trancl =
   501   {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (trancl_tac)) *}
   502   {* simple transitivity reasoner *}	    
   503 method_setup rtrancl =
   504   {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (rtrancl_tac)) *}
   505   {* simple transitivity reasoner *}
   506 
   507 *)
   508 
   509 end